Here are some of the special songs from the Jewish Prayer Service
1) Akdamut Milin - Composed by Rabbi Meir ben Yitzchak in the 11th century, in Worms, Germany, it has 90 verses in terse, difficult Aramaic which lead us through the great heights and depths of mystical understanding. It is read on Shavuot, right before the first aliyah.
2) Anim Zemirot - Song of Glory - by Rabbi Yehuda HaChassid from the 12th century. This is sung on Shabbat after musaf, with the Ark open.
3) Avinu Malkenu - Our Father, Our King - Recited on fast days and during the Days of Awe, it's origins come from the life of Rabbi Akiva [Taanis 25b] when he prayed the first five verses and brought an end to a drought. There are many variations of this beautiful meditation.
4) B'rich Shemei - Blessed is the Name - is recited after the Ark is opened and before the Torah is brought out. In Aramaic, it comes from the Zohar where it says it should be said when the Ark is opened. The section of the Zohar is Vayakhel, the parsha in the Torah when the Mishkhan is dedicated.
5) L'Chai Olamim - The Life of the Universes - is recited in the Sephard nusach on Shabbat just before Baruch She'amar.